IEB Grade 12 Analytical Geometry Cheat Sheet

Analytical geometry is about 40 marks of Paper 2. These are the formulas and proof rules you need.

Lines

• $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$; $M=\left(\frac{x_1+x_2}{2};\frac{y_1+y_2}{2}\right)$; $m=\frac{y_2-y_1}{x_2-x_1}$

• $y-y_1=m(x-x_1)$ or $y=mx+c$.

• Parallel: $m_1=m_2$. Perpendicular: $m_1{m}_2=-1$ (when both gradients exist; a vertical line is perpendicular to a horizontal one).

• $m=\tan \theta$, $0°\le \theta <180°$; add 180° to a negative calculator angle.

• Angle between lines: difference of inclinations (exterior angle of a triangle).

Circles

• Centre origin: $x^2+y^2=r^2$. Centre $(a;b)$: $(x-a{)}^2+(y-b{)}^2=r^2$.

• Expanded form: complete the square for $x$ and $y$; the right side must be positive.

• Tangent at $P$: perpendicular to radius $CP$.

• Tangent length from $T$: $\sqrt{T{C}^2-r^2}$.

• Line is a tangent if substituting gives $\Delta =0$.

Polygon proofs (minimum conditions)

• Parallelogram: diagonals share a midpoint.

• Rhombus: parallelogram with adjacent sides equal (or perpendicular diagonals).

• Rectangle: parallelogram with a right angle (or equal diagonals).

• Square: rhombus with a right angle.

• Collinear: equal gradients through a shared point.

• End with a sentence naming the property used.

Euclidean links

• Angle in a semicircle: a chord subtending 90° is a diameter.

• Line from centre to midpoint of a chord is perpendicular to it.

Know by heart

• Minimum conditions for each quadrilateral

• Completing the square for circles

On the formula sheet

• Distance, midpoint, gradient, line and circle equations, $m=\tan \theta$

Common traps

• Centre of $(x-3{)}^2+(y+2{)}^2=25$ written as $(-3;2)$.

• Radius given as $r^2$.

• Using the radius gradient as the tangent gradient.

• Negative angle of inclination.

Related topics

• lines gradient distance midpoint
• equation of a circle
• tangent to a circle
• analytical proofs of polygons